Constant Density Spanners for Wireless Ad hoc Networks

Kishore Kothapalli (JHU)Melih Onus (ASU)Christian Scheideler (JHU)Andrea Richa (ASU)

1

Ad hoc Networks

● Network created by wireless stations communicating over a wireless medium

● Two challenges– Lack of centralized infrastructure– Mobility

2

Ad hoc Networks

● Network created by wireless stations communicating over a wireless medium

● Two challenges– Lack of centralized infrastructure

– Mobility

● Need topology control in ad hoc networks– Local control strategies are needed

– Support time and energy efficient routing

● How to model ad hoc networks?– Need models that are close to reality

– But can still design algorithms using the model2

Modeling Wireless Networks

● Wireless communication very difficult to model accurately– Shape of transmission range

– Interference

– Mobility

– Physical Carrier Sensing

3

Outline Introduction

→Models of Wireless networks

● Our model

● Our results

● Problem description

– Running Example

● Conclusions

4

Models of Wireless Networks

● Unit Disk Graph (UDG)– Given a transmission radius

R, nodes u, v are connected if d(u,v) ≤ R

– Too simple model

5

uR

vu'

●What is the problem?– Transmission range could be

arbitrary shape

5

R

R

uR

vu'

u

Models of Wireless Networks

● Unit Disk Graph (UDG)– Given a transmission radius

R, nodes u, v are connected if d(u,v) ≤ R

– Too simple model

● Packet Radio Network (PRN)– Can handle arbitrary shapes

– Widely used

– Nodes u, v can communicate directly if they are within each other's transmission range, rt.

6

u

v

w

v'

Models of Wireless Networks

What is the problem?

●Model for interference too simplistic

7

u

v

w

v'

●w can still interfere at u●PRN model fails to address certain interference

problems in practice

v

n-2

s

t ≤rt

≤ rt

≤ ri

≥ rt

7

What is the problem?

u

v

w

v'

8

●Transmission Range, Interference Range– Separate values for transmission

range, interference range.– Interference range constant times

bigger than transmission range.– Used in e.g., [Adler and

Scheideler '98], [Kuhn et. al., '04]

Models of Wireless Networks

urt

vw

u'

ri

8

●Transmission Range, Interference Range– Separate values for transmission

range, interference range.– Interference range constant times

bigger than transmission range.– Used in e.g., [Adler and

Scheideler '98], [Kuhn et. al., '04]

●What is the problem?– Extension of unit disk model to

handle interference

Models of Wireless Networks

urt

vw

u'

ri

9

Outline

Introduction

Models of Wireless Networks

→Our Model

● Our results

● Problem description

– Running Example

● Conclusions

Our Model

● Transmission range, interference range via cost function

● Carrier sensing– Two types

1)Physical carrier sensing

2)Virtual carrier sensing

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Cost Function

● Gr = (V, Er), set of nodes V, Euclidean distance d(•,•)● c is a cost function on nodes

– symmetric: c(u,v) = c(v,u)

− [0,1), depends on the environment

– c(u,v) [(1-)•d(u,v), (1+)•d(u,v)]

● Edge (u,v) Er if and only if c(u,v) ≤ r

w

u

va

b

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Transmission and Interference Range

● Transmission range denoted rt(P), Interference range, r

i(P)

– If c(v,w) ≤ ri(P), node v can cause interference at node w.

– If c(v,w) ≤rt(P) then v is guaranteed to receive the message from

w provided no other node v' with c(v, v') ≤ ri(P) also transmits at

the same time.

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w

rt(P)v'

ri(P)

u

v c(v,w) rt(P)

c(v, v') ri(P)

Physical Carrier Sensing

● Provided by Clear Channel Assessment (CCA) circuit:– Monitor the medium as a function of Received Signal

Strength Indicator (RSSI)

– Energy Detection (ED) bit set to 1 if RSSI exceeds a certain threshold

– Has a register to set the threshold in dB

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Physical Carrier Sensing

● Carrier sense transmission (CST) range, denoted rst(T, P)

● Carrier sense interference (CSI) range, denoted rsi(T, P)

● Both the ranges grow monotonically in both T and P.

w

vr

st(T,P)v'

v''

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rsi(T,P) c(w,v) rst(T, P)

c(w, v') rsi(T, P)

c(w, v'') rsi(T, P)

Virtual Carrier Sensing

1. RTSst

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● Done with the use of two control signals – Request To Send (RTS)

– Clear To Send (CTS)

● DATA transmission begins after receipt of CTS

Virtual Carrier Sensing

● Done with the use of two control signals – Request To Send (RTS)

– Clear To Send (CTS)

● DATA transmission begins after receipt of CTS15

2. CTSst

Virtual Carrier Sensing

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3. DATAst

● Done with the use of two control signals – Request To Send (RTS)

– Clear To Send (CTS)

● DATA transmission begins after receipt of CTS

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Outline

Introduction

Models of Wireless Networks

Our Model

→Our Results

● Problem description

– Running Example

● Conclusions

Our Results

● More general model for ad hoc wireless networks● Constant density topological spanner for the original

network– Local-control– Self-stabilizing [Dijkstra '74]– No knowledge of size or topology of network, including

estimate of size– Nodes do not need globally distinct labels– Constant storage and constant size messages

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18

Outline

Introduction

Models of Wireless Networks

Our Model

Our Results

→Problem Description

– Running Example

● Conclusions

Topological Spanners

● Definition: Given a graph G = (V,E), find a sub-graph H = (V, E') such that d

H(u,v) ≤ t•dG(u,v)

– H is also called a t-spanner.

● Previous Work– [Alzoubi et. al., '03] 5-spanner

– [Dubhashi et. al., '03] log n – spanner

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Our Approach

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● Dominating set: Given a graph G = (V,E) a subset U such that all nodes are either in U or have a neighbor in U.– Density of U is the maximum

number of neighbors that any node has in U.

Dominator

Density = 3

● Seek a connected dominating set of constant density.

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Our Approach

● Dominating set: Given a graph G = (V,E) a subset U such that all nodes are either in U or have a neighbor in U.– Density of U is the maximum

number of neighbors that any node has in U.

Dominator

Density = 3

Our Approach

● Each round has time slots reserved for each phase of the protocol

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Time

One round

Phase I Phase II Phase III Phase I Phase II Phase III

● Three phase protocol1. Phase I: Dominating set2. Phase II: Distributed Coloring3. Phase III: Gateway Discovery

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Outline

Introduction

Models of Wireless Networks

Our Model

Our Results

Problem Description

→Running Example

● Conclusions

Phase I: Constant Density Dominating Set

● Observation: In Gr = (V, Er) any Maximal Independent Set (MIS) is also a dominating set of constant density

● Maximal Independent Set– well studied starting from [Luby '85], [Dubhashi et.

al., '03], [Kuhn et. al., '04], [Gandhi and Parthasarathy '04]

● Our solution– Uncertainties in our model make it harder– Without knowing size of network, have to use

physical carrier sensing– Randomized protocol that runs in O(log4 n) w.h.p.

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● Nodes choose a state {Active, Inactive}

● Two different sensing thresholds:

– active nodes use CSI range = rt

– inactive nodes use CST range = ri

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InactiveActive

Phase I: Constant Density Dominating Set

● If an active node sends a LEADER signal, the nodes that receive/sense the leader signal stay inactive

Changed from Active Inactive

●Nodes choose a state {Active, Inactive}● Two different sensing thresholds:

– active nodes use CSI range = rt

– inactive nodes use CST range = ri

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Inactive

Active and sent LEADER

Phase I: Constant Density Dominating Set

Phase I: Constant Density Dominating Set

DominatorsInactive nodes

● Nodes choose a state {Active, Inactive}● Two different sensing thresholds:

– active nodes use CSI range = rt

– inactive nodes use CST range = ri

● If an active node sends a LEADER signal, the nodes that receive/sense the leader signal stay inactive

● If no active node available, inactive nodes become active again

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Phase II: Distributed Coloring

● U := output of phase I, also called active nodes● Active nodes choose colors (equiv. time rounds)

such that nodes that are not riri apart have

different colors.25

ri

ri

u

vw

u

v w

Output of Phase II

● Time arranged as rounds● Each round has time slots reserved for

communication in each phase● Transmission of active node during the

corresponding round is free of interference!

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One round

Phase I Phase II Phase III Phase I Phase II Phase III

Time

Phase III: Gateway Discovery● Dominating set U may not be a connected

dominating set– Extend U by gateway nodes.

● Observation: Each node in U needs O(1) gateways.● Uses coloring achieved in Phase II to minimize

interference problems● Approach similar to that used in [Wang and Li, '03]

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Phase III Gateway Discovery

u

l

l'

CLIENT(u)

v

● If CLIENT messages interfere at active node, l, then the active nodes sends a COLLISION signal

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Phase III Gateway Discovery

ACK

● Eventually only one inactive node sends a CLIENT message to an active node

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u

l

l'v

Phase III Gateway Discovery

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●After receiving ACK from l, u advertises presence of l via ADV message.

ADV(u,l)((( )))

u

l

l'v

Phase III Gateway Discovery

Phase III

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GATEWAY(l,u,v,l')

((( )))u

l

l'v

u

l

l'v

Active node

Inactive nodeGateway nodeGateway edgeOther edges

3-Spanner

● Our construction achieves a 3-spanner of constant density for the original network.

●Total runtime = O(log4 n + (D log D) log n) whp.– D is the density of the original network

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u

l

l'v

st

Active node

Inactive nodeGateway nodeGateway edgeOther edges

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Outline

Introduction

Models of Wireless Networks

Our Model

Our Results

Problem Description

Running Example

→Conclusions

Conclusions and Future Work

● More realistic model for wireless networks● Still possible to design efficient algorithms

– Constant density 3-spanner

– Algorithms are simple and use constant sized messages, constant storage at nodes

● Further applications– Higher communication primitives e.g., broadcasting,

gathering

– Handling mobility

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